560 S. Lotfi, S.A. Zenios / European Journal of Operational Research 269 (2018) 556–576
Proposition 1. If random variable ξ has a distribution from the set
D with fixed ¯μ and
¯
, then
min
γ ∈ R
max
π∈ D
F
α
(x, γ ) = min
γ ∈ R
γ (12)
s.t. max
π∈ D
Prob { γ ≤ f (x, ξ ) } ≤ 1 − α,
and the solution is −r
f
− ( ¯μ − r
f
e )
x +
√
α
√
1 −α
x
¯
x .
Proof. From Eq. (10) in Pac and Pınar (2014) we know that
min
γ ∈ R
max
π∈ D
F
α
(x, γ ) = −r
f
− ( ¯μ − r
f
e )
x +
√
α
√
1 − α
x
¯
x . (13)
For the constraint of the optimization problem in the right-hand
side of (12) , we use Theorem 1 of Ghaoui et al. (2003) (set 1 − α
and f ( x , ξ ) instead of and −r(w, x ) , respectively), which means
max
π∈ D
Prob { γ ≤ f (x, ξ ) } ≤ 1 − α
is equivalent to
−r
f
− ( ¯μ − r
f
e )
x +
√
α
√
1 − α
x
¯
x ≤ γ .
Hence, the optimization problem in the right-hand side is equiva-
lent to
min
γ ∈ R
γ (14)
s.t. −r
f
− ( ¯μ − r
f
e )
x +
√
α
√
1 − α
x
¯
x ≤ γ ,
which has the minimum value −r
f
− ( ¯μ − r
f
e )
x +
√
α
√
1 −α
x
¯
x .
This completes the proof.
Remark
6. The left and right-hand sides of (12) are RCVaR
and RVaR, respectively, associated with the ambiguity set of
Definition 1 . Hence, the robust counterpart to VaR and CVaR op-
timization for distribution ambiguity is the same optimization
model.
Zymler, Kuhn, and Rustem (2013a,b) also derived worst-case
VaR and CVaR for piece-wise and quadratic loss functions when
there is ambiguity in distribution of asset returns but known mean
and covariance information. Both of optimization problems are ob-
tained as SDPs and they turned out to be identical for both loss
functions. When the loss function is linear (no derivative securi-
ties), the associated problems reduce to the same SOCPs as in our
Proposition 1 . Zymler et al. (2013a) extend further these identical
models to ambiguity in means and covariance matrix. They take
into account box-type ambiguity in the moment matrix but we
consider joint ellipsoidal ambiguity set for mean and covariance
matrix and obtain the robust models for a linear loss function next.
So, compared to Zymler et al. we establish the equivalence with
more general ambiguity sets when there is ambiguity in means
and covariance matrix, whereas they use more general loss func-
tions. Our result also generalizes the result of
ˇ
Cerbáková (2006)
for
the special case of symmetric distributions identified only by the
first two moments. It also establishes that the bounds obtained by
Bertsimas, Lauprete, and Samarov (2004) on VaR and CVaR for dis-
tribution ambiguity are tight.
We obtain now RVaR and RCVaR optimization models for am-
biguity in distributions, mean returns and covariance matrix.
Theorem 2. If random variable ξ has a distribution from the set D
and ( ¯μ,
¯
) ∈ U
δ
( ˆ μ,
ˆ
) . Then, the robust counterpart to VaR portfo-
lio optimization model (5) and the robust counterpart to CVaR model
(6) are both represented by the following SOCP:
min
x ∈ R
n
− r
f
− ( ˆ μ − r
f
e )
x +
max
κ∈ [0 , 1]
f (κ )
ˆ
1
2
x (15)
s.t. −
δ
√
S
ˆ
1
2
x + ( ˆ μ − r
f
e )
x ≥ d − r
f
,
where f (κ ) =
√
α
√
1 −α
(1 + δ
2(1 −κ )
S−1
+ δ
κ
S
.
Proof. See Appendix A.1 .
Remark
7. f ( κ) is a strictly concave function with lim
κ→ 0
f
(κ ) =
∞ and lim
κ→ 1
f
(κ ) = −∞ . Hence, f ( κ) has a unique maximum in
the interval (0, 1).
3.2. Constructing the ambiguity set
The ambiguity set is typically taken as input in the robust op-
timization literature. In some cases the ambiguity set can be de-
fined as the confidence regions of the statistical estimators of the
model parameters ( Goldfarb & Iyengar, 2003; Schottle & Werner,
2009 ). For other cases –such as in the applications we solve later—
we
may be given multiple estimates of model parameters which
raises the question as to what is the appropriate ambiguity set.
In finance, for instance, it is not uncommon to be given estimates
by multiple securities analysts and we need a method to construct
an ambiguity set, including its center. This issue has not been ad-
dressed in existing literature and we now propose and solve ana-
lytically a nonlinear SDP for finding the center of a joint ellipsoidal
set.
Assume K experts provide estimates for mean returns and co-
variance matrices ( ¯μ
k
,
¯
k
) , k = 1 , 2 , . . . , K. (For convenience we as-
sume they were all estimated using the same number of scenarios
S .) To construct their joint ellipsoidal ambiguity set we need to fix
the center ( ˆ μ,
ˆ
). This is obtained as the solution of a nonlinear
convex program for minimizing the l
2
-norm of the parameters δ
k
,
for k = 1 , 2 , . . . , K, where each parameter corresponds to the ellip-
soid with center ( ˆ μ,
ˆ
) containing observation ( ¯μ
k
,
¯
k
) . Referring
to Definition 2 the optimization problem is given by
min
ˆ μ∈ R
n
,
ˆ
∈ S
n
++
K
k =1
S ( ¯μ
k
− ˆ μ)
ˆ
−1
( ¯μ
k
− ˆ μ) +
S − 1
2
ˆ
−
1
2
(
¯
k
−
ˆ
)
ˆ
−
1
2
2
tr
,
(16)
where S
n
++
is the set of all n -dimensional, symmetric, positive def-
inite matrices. This problem is equivalent to
min
ˆ μ∈ R
n
,
ˆ
∈ S
n
++
K
k =1
S ( ¯μ
k
− ˆ μ)
ˆ
−1
( ¯μ
k
− ˆ μ) +
S − 1
2
ˆ
−
1
2
(
¯
k
−
ˆ
)
ˆ
−
1
2
2
tr
.
(17)
The next theorem gives the solution of this problem, if a solu-
tion exists.
Theorem 3. If (17) is solvable, then it admits the following solution:
1. ˆ μ =
1
K
K
k =1
¯μ.